The sum of a maximally monotone linear relation and the subdifferential of a proper lower semicontinuous convex function is maximally monotone

TL;DR

This paper proves that the sum of a maximally monotone linear relation and a subdifferential of a proper lower semicontinuous convex function is maximally monotone under certain domain conditions, advancing the understanding of maximal monotonicity in Monotone Operator Theory.

Contribution

It establishes the maximal monotonicity of the sum $A + abla f$ when $A$ is a maximally monotone linear relation and $f$ is a proper lower semicontinuous convex function with specific domain intersections, filling a key gap in the theory.

Findings

Proves maximal monotonicity of $A + abla f$ under domain conditions.

Shows $A + abla f$ is of type (FPV).

Complements existing results by Verona and Verona.

Abstract

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar's constraint qualification holds. In this paper, we prove the maximal monotonicity of $A+\partial f$ provided that $A$ is a maximally monotone linear relation, and $f$ is a proper lower semicontinuous convex function satisfying $\dom A\cap\inte\dom \partial f\neq\varnothing$. Moreover, $A+\partial f$ is of type (FPV). The maximal monotonicity of $A+\partial f$ when $\intdom A\cap\dom \partial f\neq\varnothing$ follows from a result by Verona and Verona, which the present work complements.